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We give accurate estimates for the constants K(A(I),n,x)=supf∈A(I)|Lnf(x)−f(x)|ωσ2(f;1/n),x∈I,n=1,2,…,$$ K\bigl(\mathcal{A}(I), n, x\bigr)=\sup_{f\in\mathcal{A}(I)}\frac{|L_{n} f(x)-f(x)|}{\omega_{\sigma}^{2} ( f; 1/\sqrt{n} )}, \quad x\in I, n=1,2,\ldots, $$ where I=R$I=\mathbb {R}$ or I=[0,∞)$I=[0,\infty)$, Ln$L_{n}$ is a positive linear operator acting on real functions f defined on the interval I, A(I)$\mathcal{A}(I)$ is a certain subset of such function, and ωσ2(f;⋅)$\omega _{\sigma}^{2} (f;\cdot)$ is the Ditzian-Totik modulus of smoothness of f with weight function σ. This is done under the assumption that σ is concave and satisfies some simple boundary conditions at the endpoint of I, if any. Two illustrative examples closely connected are discussed, namely, Weierstrass and Szàsz-Mirakyan operators. In the first case, which involves the usual second modulus, we obtain the exact constants when A(R)$\mathcal{A}(\mathbb {R})$ is the set of convex functions or a suitable set of continuous piecewise linear functions.

Towards the best constant in front of the Ditzian-Totik modulus of smoothness